Page 203 - Modern Control Systems

Section 3.4 Signal-Flow Graph and Block Diagram Models 177
The block diagram model can also be constructed directly from Equation (3.46).
Define the intermediate variable Z{s) and rewrite Equation (3.46) as

= 1^1 V 2 Z{s)
G(s) = + b 2s + b lS + b 0
U(s) S A + a 3s 3 + a 2s 2 + a xs + a 0 Z(s)'
Notice that, by multiplying by Z(s)/Z(s), we do not change the transfer function,
G(s). Equating the numerator and denominator polynomials yields
3
Y(s) = [b 3s + b 2s 2 + b iS + b Q]Z(s)
and
4
U(s) = [s + a 3s 3 + a 2s 2 + ais + a Q]Z(s).
Taking the inverse Laplace transform of both equations yields the differential
equations
l
2
ud z d z dz
y = b + b 2 + bl + b z
^ ^ * "
and
4 3 2
d z d z d z dz
4 3 3 2 2 l
dt dt dt dt
Define the four state variables as follows:
x x-z

X 2 = X\ = Z
x
3 — X 2 — Z
=
Xq — X 3 Z-
Then the differential equation can be written equivalently as
xi = x 2,

X 2 = X 3,
X3 = X4,
and

X\ = Cl()X\ Cl\X 2 #2-^-3 ^3-^4 "^ Mi
and the corresponding output equation is

y = b 0x { + b xx 2 + b 2x 3 + b 3x 4.
The block diagram model can be readily obtained from the four first-order differential
equations and the output equation as illustrated in Figure 3.11(b).
Furthermore, the output is simply

y(t) = b 0x x + b xx 2 + b 2x 3 + b 3x 4. (3.48)